Our focus has been on the asymptotic performance of digital quantum simulation throughout this paper. However, we believe that the techniques we have developed can also be used to give quantum simulations with low constant-prefactor overhead, for instance, through a more careful application of our [prop:multilayer]Proposition prop:multilayer, [prop:multilayerseminorm]Proposition prop:multilayerseminorm, [prop:pathboundseminorm]Proposition prop:pathboundseminorm and [prop:cntpathsparse]Proposition prop:cntpathsparse. Such improvements would especially benefit the simulation of plane-wave-basis electronic structure, where many pairs of Hamiltonian terms commute and the number of electrons can be significantly smaller than the spin-orbital number. Existing numerical studies almost exclusively used the second-order Suzuki formula [1]}, [2]}, [3]}, [4]}, [5]} and their results did not fully leverage the commutativity of the Hamiltonian and the initial-state knowledge, which may then be improved by the techniques presented here using general Trotterization schemes.
